Subgroups — C₁₃×#22440 #15708779

Order 3328.

1386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#441	C₂	1	2	yes	maximal, complemented
3	Q₈×C₁₃×C₂²⋊C₄	C₂	1	2	yes	maximal
4	C₁₃×#1971	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1878	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1807	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1787	C₂	1	2	yes	maximal
8	C₁₃×#1752	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
10	C₂²×C₁₃×Q₁₆	—	2	8	no	—
11	C₂²×C₁₃×Q₁₆	C₄	1	8	yes	complemented
12	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
13	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
17	C₁₃×#441	—	2	8	no	—
18	C₂×C₁₃×#106	C₂²	1	2	yes	—
19	C₂×C₁₃×#103	C₂²	1	4	yes	—
20	C₂×C₁₃×#103	C₂²	1	4	yes	—
21	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×#97	C₂²	1	2	yes	—
23	C₁₃×#391	—	2	4	no	—
24	C₁₃×#391	C₂²	1	4	yes	—
25	C₁₃×#380	—	2	2	no	—
26	C₁₃×#380	C₂²	1	2	yes	—
27	C₄×C₁₃×C₂²⋊C₄	—	2	2	no	—
28	C₄×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₁₃×#340	C₂²	1	2	yes	—
30	C₁₃×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₁₃	—	2	2	no	—
32	C₂²×Q₈×C₁₃	—	2	4	no	—
33	C₂²×Q₈×C₁₃	—	2	8	no	—
34	C₂²×Q₈×C₁₃	—	4	4	no	—
35	C₂²×Q₈×C₁₃	C₂×C₄	1	8	yes	—
36	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
38	C₂³×C₄×C₁₃	—	2	2	no	—
39	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₁₃×Q₁₆	—	2	8	no	—
41	C₂×C₁₃×Q₁₆	—	2	32	no	—
42	C₂×C₁₃×Q₁₆	—	2	4	no	—
43	C₂×C₁₃×Q₁₆	C₂×C₄	1	8	yes	—
44	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
45	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
46	C₂×C₁₃×Q₁₆	D₄	1	8	yes	—
47	C₂²×C₈×C₁₃	—	2	4	no	—
48	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₄×Q₈×C₁₃	—	2	8	no	—
53	C₄×Q₈×C₁₃	—	4	4	no	—
54	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
55	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
56	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
57	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
58	C₂×C₄²×C₁₃	—	2	4	no	—
59	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
60	C₁₃×#106	—	2	4	no	—
61	C₁₃×#103	—	2	8	no	—
62	C₁₃×#103	—	4	4	no	—
63	C₁₃×C₂²⋊C₈	—	2	2	no	—
64	C₁₃×#97	—	2	4	no	—
65	C₁₃×#95	—	2	4	no	—
66	C₁₃×#95	—	2	4	no	—
67	C₁₃×#95	—	2	2	no	—
68	C₁₃×#95	D₄	1	2	yes	—
69	#22440	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×Q₈×C₁₃	—	2	8	no	—
72	C₂×Q₈×C₁₃	—	2	2	no	—
73	C₂×Q₈×C₁₃	—	2	2	no	—
74	C₂×Q₈×C₁₃	—	2	4	no	—
75	C₂×Q₈×C₁₃	—	2	16	no	—
76	C₂×Q₈×C₁₃	—	2	4	no	—
77	C₂×Q₈×C₁₃	—	2	4	no	—
78	C₂×Q₈×C₁₃	—	4	8	no	—
79	C₂×Q₈×C₁₃	—	4	8	no	—
80	C₂×Q₈×C₁₃	—	4	16	no	—
81	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
82	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
83	C₂×Q₈×C₁₃	C₂×D₄	1	8	yes	—
84	C₂²×C₄×C₁₃	—	2	2	no	—
85	C₂²×C₄×C₁₃	—	2	2	no	—
86	C₂²×C₄×C₁₃	—	2	2	no	—
87	C₂²×C₄×C₁₃	—	2	8	no	—
88	C₂²×C₄×C₁₃	—	2	2	no	—
89	C₂²×C₄×C₁₃	—	2	4	no	—
90	C₂²×C₄×C₁₃	—	4	8	no	—
91	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
93	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	#41	1	2	yes	—
95	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₁₃×Q₁₆	—	2	16	no	—
97	C₁₃×Q₁₆	—	2	8	no	—
98	C₁₃×Q₁₆	C₂²⋊C₄	1	16	yes	complemented
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	—	2	8	no	—
101	C₂×C₈×C₁₃	—	2	2	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
107	C₁₃×C₄⋊C₄	—	2	4	no	—
108	C₁₃×C₄⋊C₄	—	2	4	no	—
109	C₁₃×C₄⋊C₄	—	4	8	no	—
110	C₁₃×C₂²⋊C₄	—	2	4	no	—
111	C₁₃×C₂²⋊C₄	—	4	4	no	—
112	C₄²×C₁₃	—	2	2	no	—
113	C₄²×C₁₃	—	2	2	no	—
114	C₄²×C₁₃	—	4	8	no	—
115	C₂³×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
116	C₂×#441	C₂×C₁₃	1	2	yes	complemented
117	Q₈×C₂²⋊C₄	C₂×C₁₃	1	2	yes	—
118	#1971	C₂×C₁₃	1	2	yes	complemented
119	#1878	C₂×C₁₃	1	4	yes	complemented
120	#1807	C₂×C₁₃	1	1	yes	characteristic
121	#1787	C₂×C₁₃	1	2	yes	—
122	#1752	C₂×C₁₃	1	1	yes	characteristic
123	C₂³×C₁₃	—	2	2	no	—
124	C₂³×C₁₃	—	2	2	no	—
125	C₂³×C₁₃	C₄×D₄	1	2	yes	—
126	C₂³×C₁₃	#137	1	4	yes	—
127	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	Q₈×C₁₃	—	2	4	no	—
129	Q₈×C₁₃	—	2	8	no	—
130	Q₈×C₁₃	—	2	4	no	—
131	Q₈×C₁₃	—	2	4	no	—
132	Q₈×C₁₃	—	4	16	no	—
133	Q₈×C₁₃	—	4	4	no	—
134	Q₈×C₁₃	C₂×C₂²⋊C₄	1	8	yes	—
135	C₂×C₄×C₁₃	—	2	4	no	—
136	C₂×C₄×C₁₃	—	2	8	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	4	no	—
141	C₂×C₄×C₁₃	—	2	4	no	—
142	C₂×C₄×C₁₃	—	2	4	no	—
143	C₂×C₄×C₁₃	—	2	4	no	—
144	C₂×C₄×C₁₃	—	2	4	no	—
145	C₂×C₄×C₁₃	—	4	16	no	—
146	C₂×C₄×C₁₃	—	4	4	no	—
147	C₂×C₄×C₁₃	—	4	4	no	—
148	C₂×C₄×C₁₃	—	4	4	no	—
149	C₂×C₄×C₁₃	—	4	4	no	—
150	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
151	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
153	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
155	C₈×C₁₃	—	2	2	no	—
156	C₈×C₁₃	—	4	2	no	—
157	C₈×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
158	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
159	C₂²×Q₁₆	—	2	8	no	—
160	C₂²×Q₁₆	C₄×C₁₃	1	8	yes	complemented
161	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
162	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
163	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
164	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
165	C₂×C₄×Q₈	C₂²×C₁₃	1	4	yes	—
166	#441	—	2	8	no	—
167	C₂×#106	C₂²×C₁₃	1	2	yes	—
168	C₂×#103	C₂²×C₁₃	1	4	yes	—
169	C₂×#103	C₂²×C₁₃	1	4	yes	—
170	C₂×C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
171	C₂×#97	C₂²×C₁₃	1	2	yes	—
172	#391	—	2	4	no	—
173	#391	C₂²×C₁₃	1	4	yes	—
174	#380	—	2	2	no	—
175	#380	C₂²×C₁₃	1	2	yes	—
176	C₄×C₂²⋊C₄	—	2	2	no	—
177	C₄×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
178	#340	C₂²×C₁₃	1	2	yes	—
179	#336	C₂²×C₁₃	1	2	yes	—
180	C₂²×C₁₃	—	2	2	no	—
181	C₂²×C₁₃	—	2	2	no	—
182	C₂²×C₁₃	—	2	8	no	—
183	C₂²×C₁₃	#441	1	4	yes	complemented
184	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
185	C₂²×C₁₃	D₄²	1	2	yes	—
186	C₂²×C₁₃	C₂×#137	1	2	yes	—
187	C₂²×C₁₃	#578	1	2	yes	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	8	no	—
190	C₄×C₁₃	—	4	4	no	—
191	C₄×C₁₃	—	4	4	no	—
192	C₄×C₁₃	—	8	4	no	—
193	C₄×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
194	C₄×C₁₃	#518	1	1	yes	characteristic
195	C₄×C₁₃	#546	1	2	yes	—
196	C₂²×Q₈	—	2	2	no	—
197	C₂²×Q₈	—	2	4	no	—
198	C₂²×Q₈	—	2	8	no	—
199	C₂²×Q₈	—	4	4	no	—
200	C₂²×Q₈	C₂×C₄×C₁₃	1	8	yes	—
201	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
202	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
203	C₂³×C₄	—	2	2	no	—
204	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
205	C₂×Q₁₆	—	2	8	no	—
206	C₂×Q₁₆	—	2	32	no	—
207	C₂×Q₁₆	—	2	4	no	—
208	C₂×Q₁₆	C₂×C₄×C₁₃	1	8	yes	—
209	C₂×Q₁₆	C₂×C₄×C₁₃	1	4	yes	—
210	C₂×Q₁₆	C₂×C₄×C₁₃	1	4	yes	—
211	C₂×Q₁₆	D₄×C₁₃	1	8	yes	—
212	C₂²×C₈	—	2	4	no	—
213	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
214	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
215	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
217	C₄×Q₈	—	2	8	no	—
218	C₄×Q₈	—	4	4	no	—
219	C₂×C₄⋊C₄	—	2	4	no	—
220	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
221	C₂×C₂²⋊C₄	—	2	2	no	—
222	C₂×C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
223	C₂×C₄²	—	2	4	no	—
224	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
225	#106	—	2	4	no	—
226	#103	—	2	8	no	—
227	#103	—	4	4	no	—
228	C₂²⋊C₈	—	2	2	no	—
229	#97	—	2	4	no	—
230	#95	—	2	4	no	—
231	#95	—	2	4	no	—
232	#95	—	2	2	no	—
233	#95	D₄×C₁₃	1	2	yes	—
234	C₂×C₁₃	—	2	4	no	—
235	C₂×C₁₃	D₄×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₁₃	C₂×#441	1	1	yes	characteristic
237	C₂×C₁₃	#2950	1	1	yes	characteristic
238	C₂×C₁₃	#3270	1	4	yes	—
239	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₂×Q₈	—	2	8	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	2	16	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	4	8	no	—
248	C₂×Q₈	—	4	8	no	—
249	C₂×Q₈	—	4	16	no	—
250	C₂×Q₈	C₂²×C₄×C₁₃	1	4	yes	—
251	C₂×Q₈	C₂²×C₄×C₁₃	1	4	yes	—
252	C₂×Q₈	C₂×D₄×C₁₃	1	8	yes	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	8	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	4	8	no	—
260	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
261	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
262	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₂²×C₄	C₁₃×#41	1	2	yes	—
264	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
265	Q₁₆	—	2	16	no	—
266	Q₁₆	—	2	8	no	—
267	Q₁₆	C₁₃×C₂²⋊C₄	1	16	yes	complemented
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	8	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
274	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
275	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	4	8	no	—
279	C₂²⋊C₄	—	2	4	no	—
280	C₂²⋊C₄	—	4	4	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	4	8	no	—
284	C₁₃	#22440	1	1	yes	characteristic, direct product factor, Sylow
285	C₂³	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	C₄×D₄×C₁₃	1	2	yes	—
288	C₂³	C₁₃×#137	1	4	yes	—
289	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
290	Q₈	—	2	4	no	—
291	Q₈	—	2	8	no	—
292	Q₈	—	2	4	no	—
293	Q₈	—	2	4	no	—
294	Q₈	—	4	16	no	—
295	Q₈	—	4	4	no	—
296	Q₈	C₂×C₁₃×C₂²⋊C₄	1	8	yes	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	4	16	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
313	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
315	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
316	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
317	C₈	—	2	2	no	—
318	C₈	—	4	2	no	—
319	C₈	C₂×C₁₃×C₂²⋊C₄	1	4	yes	—
320	C₂²	—	2	8	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	C₁₃×#441	1	4	yes	complemented
324	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
325	C₂²	D₄²×C₁₃	1	2	yes	—
326	C₂²	C₂×C₁₃×#137	1	2	yes	—
327	C₂²	C₁₃×#578	1	2	yes	—
328	C₄	—	2	2	no	—
329	C₄	—	2	8	no	—
330	C₄	—	4	4	no	—
331	C₄	—	4	4	no	—
332	C₄	—	8	4	no	—
333	C₄	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
334	C₄	C₁₃×#518	1	1	yes	characteristic
335	C₄	C₁₃×#546	1	2	yes	—
336	C₂	—	2	4	no	—
337	C₂	D₄×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
338	C₂	C₂×C₁₃×#441	1	1	yes	characteristic
339	C₂	C₁₃×#2950	1	1	yes	characteristic
340	C₂	C₁₃×#3270	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄×C₁₃×C₂²⋊C₄ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₁₃×#441	C₂	semidirect
4. C₁₃×#1971	C₂	semidirect
5. C₁₃×#1878	C₂	semidirect
11. C₂²×C₁₃×Q₁₆	C₄	semidirect
69. #22440	C₁₃	direct (N × H)
98. C₁₃×Q₁₆	C₂²⋊C₄	semidirect
116. C₂×#441	C₂×C₁₃	semidirect
118. #1971	C₂×C₁₃	semidirect
119. #1878	C₂×C₁₃	semidirect
160. C₂²×Q₁₆	C₄×C₁₃	semidirect
183. C₂²×C₁₃	#441	semidirect
267. Q₁₆	C₁₃×C₂²⋊C₄	semidirect
284. C₁₃	#22440	direct (N × H)
323. C₂²	C₁₃×#441	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₁₃×Q₁₆	2
C₂×C₁₃×#441	2
Q₈×C₁₃×C₂²⋊C₄	2
C₁₃×#1971	2
C₁₃×#1878	2
C₁₃×#1807	2
C₁₃×#1787	2
C₁₃×#1752	2
#22440	13

Subgroups — C13×#22440 #15708779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#22440 #15708779